CHAP_3_SEC1.PDF

8/14/2019 CHAP_3_SEC1.PDF

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57:020 Mechanics of Fluids and Transport Processes Chapter 3Professor Fred Stern Typed by Stephanie Schrader Fall 1999 1

Chapter 3: Fluid Statics

3.1 Pressure

For a static fluid, the only stress is the normal stress since by definition a fluid subjected to a shear stress must deformand undergo motion. Normal stresses are referred to as

pressure p.

For the general case, the stress on a fluid element or at a point is a tensor

For a static fluid,ij = 0 i j i.e., shear stresses = 0

ii = p = xx = yy = zz i = j i.e., normal stresses= p

Also shows that p is isotropic i.e., one value at a pointwhich is independent of direction, a scalar.

ij = stress tensor

= xx xy xzyx yy yz

zx zy zz

i = force= direction


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x z

Definition of Pressure:

dA

dF

A

F

0A

lim p =

=

F = normal force acting over A N/m 2 = Pa (Pascal)

As already noted, p is a scalar, which can be easilydemonstrated by considering the equilibrium of forces on awedge-shaped fluid element

GeometryA = lyx = lcos z = lsin

Fx = 0 pnA sin - p xA sin = 0 pn = p x

Fz = 0-pnA cos + p zA cos - W = 0

y)sin)(cos(2

W = ll

pnA

pxAsin

W=wei ht pzAcos

x

l

z

W = mg= Vg= V

V = xzy


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+ = p pn z 2 0l sin

p p for n z= l 0 i.e., p n = p x = p y = p z

p is single valued at a point and independent of direction

A body/surface in contact with a static fluid experiences aforce due to p

=BS

p dAn pF

Note: if p = constant, F p = 0 for a closed body

Scalar form of Greene's Theorem:f nds fdV

s V =

f = constant f = 0


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Pressure Transmission

Pascal's law: in a closed system, a pressure change

produced at one point in the system is transmittedthroughout the entire system.

Absolute Pressure, Gage Pressure, and Vacuum

For p A>p a, pg = p A p a = gage pressure

For p A

0

pA > p a pa = atmospheric

pressure =101.325 kPa

pA = 0 = absolutezero


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3.2 Pressure Variation with Elevation

Basic Differential Equation

For a static fluid, pressure varies only with elevation withinthe fluid. This can be shown by consideration ofequilibrium of forces on a fluid element

Newton's law (momentum principle) applied to a staticfluid

F = ma = 0 for a static fluidi.e., Fx = Fy = Fz = 0

Fz = 0

pdxdy p pz dz dxdy gdxdydz + =( )

0

pz

g= =

Basic equation for pressure variation with elevation

1st order Taylor series

estimate for pressurevariation over dz


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0y

p

0dxdz)dyy

p p( pdxdz

0Fy

=

=+

=

0x

p

0dydz)dxx p

p( pdydz

0Fx

=

=+

=

For a static fluid, the pressure only varies with elevation zand is constant in horizontal xy planes.

The basic equation for pressure variation with elevation can be integrated depending on whether = constant or = (z), i.e., whether the fluid is incompressible (liquid orlow-speed gas) or compressible (high-speed gas) sinceg constant

Pressure Variation for a Uniform-Density Fluid= g = constant

dpdz

=

dp = dz p = z + constant p + z = constant

piezometric pressure


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Alternate forms:

p1 + z1 = p 2 + z2

p = - z


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Pressure Variation for Compressible Fluids:

Basic equation for pressure variation with elevation

g)z, p(dzdp = = =

Pressure variation equation can be integrated for (p,z)known. For example, here we solve for the pressure in theatmosphere assuming (p,T) given from ideal gas law, T(z)known, and g g(z).

p = RT R = gas constant = 287 J/kg K p,T in absolute scale

RT pg

dzdp =

)z(Tdz

R g

pdp

= which can be integrated for T(z) known

dry air


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zo = earth surface= 0

po = 101.3 kPa

T = 15C

= 6.5 K/km

Pressure Variation in the Troposphere

T = T o (z z o) linear decrease

To = T(z o) where p = p o(zo) known = lapse rate = 6.5 K/km

)]zz(T[dz

R g

pdp

oo =

dz'dz

)zz(T'z oo=

=

constant)]zz(Tln[R g pln oo +=

use reference condition

constantTlnR g

pln oo +=

solve for constant

R g

o

oo

o

o

oo

o

T)zz(T

p p

T)zz(T

lnR

g p p

ln

=

=

i.e., p decreases for increasing z


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Pressure Variation in the Stratosphere

T = T s = 55C

dp p

gR

dzTs

=

constantzRT

g pln

s+=

use reference condition to find constant

]RT/g)zz(exp[ p p

e p p

soo

RT/g)zz(

o

s0

=

=

i.e., p decreases exponentially for increasing z.

See examples 3.5 and 3.6 in text for p analysis inTroposphere and Stratosphere, respectively.


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3.3 Pressure Measurements

Pressure is an important variable in fluid mechanics and

many instruments have been devised for its measurement.Many devices are based on hydrostatics such as barometersand manometers, i.e., determine pressure throughmeasurement of a column (or columns) of a liquid usingthe pressure variation with elevation equation for anincompressible fluid.

More modern devices include Bourdon -Tube Gage(mechanical device based on deflection of a spring) and

pressure transducers (based on deflection of a flexiblediaphragm/membrane). The deflection can be monitored

by a strain gage such that voltage output is p acrossdiaphragm, which enables electronic data acquisition withcomputers.

In this course we will use both manometers and pressuretransducers in EFD labs 2 and 3.


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Manometry

1. Barometer

pv + Hgh = p atm

i.e., p atm = Hgh p v 0 i.e., vapor pressure Hgnearly zero at normal T

h 76 cm patm 101 kPa (or 14.6 psia)

Note: p atm is relative to absolute zero, i.e., absolute pressure. p atm = p atm(location, weather)

Consider why water barometer is impracticalOHOHHgHg 22 hh =

.ft34cm6.1033766.13hShh HgHgHgOH

HgOH

2

2==== =


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patm

2. Piezometer

patm + h = p pipe = p absolute

p = h gage

Simple but impractical for large p and vacuum pressures(i.e., p abs < p atm)

3. U-tube or differential manometer

p1 + mh l = p 4 p1 = p atm p4 = mh l gage

= w[Smh S l ]

for gases S


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Air at 20 C is in pipe with a water manometer. For givenconditions compute gage pressure in pipe.

l = 140 cmh = 70 cm

p4 = ? gage (i.e., p 1 = 0)

p1 + h = p 3 step-by-step method p3 - air l = p 4 p1 + h - air l = p 4 complete circuit method

h - air l = p 4 gage

water (20C) = 9790 N/m 3 p3 = h = 6853 Pa [N/m 2] air = g

pabs( )( )

3atm3 m/kg286.1)27320(287

1013006853273CR

p pRT p =+

+=++==

K air = 1.286 9.81m/s 2 = 12.62 N/m 3

note air


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A differential manometer determines the difference in pressures at two points and when the actual pressureat any point in the system cannot be determined.

h)()( p p

p)h(h p

f m12f 21

22f m1f 1

+ =

=+

ll

ll

h1 p pf

m2

f

21

f

1

=

+

+

ll

difference in piezometric head

if fluid is a gas f

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